Tissot's indicatrix helps illustrate map projection distortion

Tissot’s indicatrix was first developed by French mathematician Nicolas Auguste Tissot in 1859 as a tool to characterize the distortion due to projecting a spherical representation of the earth onto a flat surface, i.e., a map projection.

This graphic tool helps you see what type of distortion occurs in the size, shape, and orientation of ellipses that would be circles were they located on the earth’s surface. When a circle is projected onto a flat surface through the map projection process, the result is an ellipse whose semi-minor and semi-major axes indicate the two principal directions (often but not always north-south and east-west) along which scale is maximal and minimal at that point on the map.

A single ellipse is called an “indicatrix”, and it shows the distortion at the point where it is centered. Because scale distortion varies across a map, usually Tissot’s indicatrices (the plural of indicatrix) are repeated across a map at regular intervals to illustrate the spatial pattern of distortion. It is common to place them at the intersections of parallels and meridians. In the above example, indicatrices are placed at the intersections of the 30 degree parallels and meridians that make up the graticule between 60 degrees north and south.

Figure 1. Tissot’s indicatrices are used to show the pattern of distortion due to the use of this WGS84 geographic (Plate Carree) projection.

Tissot’s indicatrix is a valuable tool in understanding and teaching about map projections, both to illustrate linear, angular, and areal distortion and to show graphically the calculations of the magnitude of distortion at each point.

On maps that use a conformal projection, in which each point preserves angles projected from the geometric model, the Tissot’s indicatrices are all circles. Because it is not possible to preserve angles and shapes at the same time, the apparent sizes of the ellipses will vary across the map.

Figure 2. On this map with the Mercator projection, all indicatrices are circles because a conformal projection is used.

On maps that use an equal area projection, in which the property of equivalence is preserved, the relative sizes of the indicatrices are retained (figure 3). But equal area projections cannot also be conformal, so the ellipses will be distorted in shape and orientation across the map.

Figure 3. On this map with the Mollweide projection, all indicatrices are apparently equal in size because an equal area projection is used.

For other map projections, such as compromise projections, which are neither equal area nor compromise, the area, shape, and orientation of the indicatrices may vary across the map (figure 4).

Figure 4. On this map with the Winkel Tripel projection, the indicatrices vary in size, shape, and orientation because a compromise projection is used.

You can download this map template to see for yourself how different map projections produce different patterns of distortion. Simply open the map document and change the map projection of the data frame. To use a different projection, right click the data frame name (“Layers”) in the Table of Contents, click Properties, and then click the Coordinate System tab. Opening the Predefined > Projected Coordinate Systems > World folder, you can select from a wide variety of map projections and modify them as you wish. Watch the indicatrices to see how the patterns of distortion change with each modification you make.

By the way, can anyone guess which map projection was used to make the thumbnail for this blog entry?

Thanks to Jon Kimerling, Professor Emeritus at Oregon State University and primary author of Map Use, Sixth Edition, and Melita Kennedy, Senior Product Engineer for Map Projections and Transformations, for their help with this blog entry.

I was planning on using this example, among others, to explain distortion and projections to my co-workers. Sadly we are stuck at version 9.3.1 for awhile. Any chance you could save it down to that version.

Congratulations, knwin! You are correct! I used the Hotine world projection and did not change any of the default parameters. Hotine is an oblique Mercator projection – people sometimes differentiate between Hotine and oblique Mercator based on what parameters are supported (because the underlying algorithms have differences). Congratulations again!

Congratulations, knwin! You are correct! I used the Hotine world projection and did not change any of the default parameters. Hotine is an oblique Mercator projection – people sometimes differentiate between Hotine and oblique Mercator based on what parameters are supported (because the underlying algorithms have differences). Congratulations again!

Very useful post Aileen, and thanks for providing the map doc to play with. There’s still one thing that I can’t quite get my head around though. If I use an equal-area projection, and slide a coin around on the surface, will the area of map covered by that coin represent a constant geographic area? Or would my coin have to change shape as I slide it around to match the changing shape of the indicatrices, in order to cover a constant area? I guess a more relevant way to phrase the question is: if I project a raster to an appropriate equal-area projection, will each pixel represent the same geographic area? Seems like a fairly silly question, given that that’s what I would assume equal area grids are designed to do… but there’s something about the changing shape of the indicatrices that’s throwing me off (although I understand that projections cannot maintain both area and shape). Can you shed any light on this for me?

Response to John’s comment form David Burrows, one of our Esri map projections gurus:

John’s assumption is, in fact, correct that a shape of a given size on an equal area projection slid across the actual projected map should represent the same area on the ground. It is just that on the actual ground, that shape, say a square or a circle, will no longer be a square or a circle.

The opposite is also true. A square or a circle on the ground of a given area will not necessarily remain a square or a circle on the projected equal area map.

Both of these issues are related to the distortion of shapes that is necessary in order to retain the equal area characteristic.

Some confusion might have come from this question from John’s in a previous email –

“If I use an equal-area projection, and slide a coin around on the surface,
will the area of map covered by that coin represent a constant geographic area?”

Well, slide a coin around on the surface of WHAT? The map or the earth?

Let’s rephrase the question –
Given a map in which an equal area projection was used, if I slide a coin around on the map, will the actual ground area on the ellipsoid represented on the map under the coin remain constant wherever I place the coin?

The answer is yes. But, the shape does not remain the same between the map and the surface of the ellipsoid. So, the circle on the map is not nessarliy a circular area on the ground.

And now, on to your raster problem.

The easiest thing for you to do would be to use a Cylindrical Equal Area projection.
Why? Because it is rectangular and lends itself well to square rasters fitting nicely within the boundaries. Every raster cell will represent the same area on the ground.
So a grid of raster cells, let’s say each cell 100 by 100 meters, will be a hectare on the ground (10000 sq meters). Of course, your scale factor on the projected coordinate system would be 1.0. Cells 50 by 50 meters would be 1/4 hectare. Etc, etc…

The added feature of the Cylindrical Equal area is that the cell boundaries in your raster would run along latitude and longitude lines, therefore each cell would be a lune section.
The actual longitude radial “width” would be constant, but the latitude span would change per cell depending on the latitude of placement. If you think about it, it makes sense.
As actual ground distance between longitudes decreases as you move towards the poles, you have to increase your latitude coverage in order to retain equal areas. This is the basis for the concept of “authalic latitude” and the math for the calculation of lune section area.